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Multi-skewed Brownian motion and diffusion in layered media

Author: Jorge M. Ramirez
Journal: Proc. Amer. Math. Soc. 139 (2011), 3739-3752
MSC (2010): Primary 60J60; Secondary 60G17
Published electronically: March 3, 2011
MathSciNet review: 2813404
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Abstract: Multi-skewed Brownian motion $ B^{\alpha} = \{B^{\alpha}_t: t \geq 0\}$ with skewness sequence $ \alpha = \{\alpha_{k}: k\in \mathbb{Z}\}$ and interface set $ S= \{x_{k}: k \in \mathbb{Z}\}$ is the solution to $ X_t = X_0 + B_t + \int_{\mathbb{R}} L^X(t,x) {d} \mu(x)$ with $ \mu = \sum_{k \in \mathbb{Z}}(2 \alpha_{k}-1) \delta_{x_{k}}$. We assume that $ \alpha_{k} \in (0,1) \setminus \{\frac{1}{2}\}$ and that $ S$ has no accumulation points. The process $ B^{\alpha}$ generalizes skew Brownian motion to the case of an infinite set of interfaces. Namely, the paths of $ B^{\alpha}$ behave like Brownian motion in $ \mathbb{R} \smallsetminus S$, and on $ B^{\alpha}_0 = x_k$, the probability of reaching $ x_k+\delta$ before $ x_k-\delta$ is $ \alpha_k$, for any $ \delta$ small enough, and $ k \in \mathbb{Z}$. In this paper, a thorough analysis of the structure of $ B^{\alpha}$ is undertaken, including the characterization of its infinitesimal generator and conditions for recurrence and positive recurrence. The associated Dirichlet form is used to relate $ B^{\alpha}$ to a diffusion process with piecewise constant diffusion coefficient. As an application, we compute the asymptotic behavior of a diffusion process corresponding to a parabolic partial differential equation in a two-dimensional periodic layered geometry.

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  • 1. T. A. Appuhamillage, V. A. Bokil, E. Thomann, Edward C. Waymire, and B. D. Wood, Occupation and local times for skew Brownian motion with applications to dispersion across an interface, Annals of Applied Probability 21 (2011), no. 1, 183-214.
  • 2. -, Solute transport across an interface: A Fickian theory for skewness in breakthrough curves, Water Resources Research 46 (2010), W07511.
  • 3. Martin Barlow, Jim Pitman, and Marc Yor, Une extension multidimensionnelle de la loi de l’arc sinus, Séminaire de Probabilités, XXIII, Lecture Notes in Math., vol. 1372, Springer, Berlin, 1989, pp. 294–314 (French). MR 1022918,
  • 4. Richard F. Bass and Zhen-Qing Chen, One-dimensional stochastic differential equations with singular and degenerate coefficients, Sankhyā 67 (2005), no. 1, 19–45. MR 2203887
  • 5. C. Berentsen, M. Verlaan, and C. Kruijsdijk, Upscaling and reversibility of Taylor dispersion in heterogeneous porous media, Physical Review E 71 (2005), 046308-1, 046308-16.
  • 6. B. Berkowitz, A. Cortis, I. Dror, and H. Scher, Laboratory experiments on dispersive transport across interfaces: The role of flow direction, Water Resources Research 45 (2009).
  • 7. Rabi N. Bhattacharya and Edward C. Waymire, Stochastic processes with applications, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1990. A Wiley-Interscience Publication. MR 1054645
  • 8. Leo Breiman, Probability, Classics in Applied Mathematics, vol. 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. Corrected reprint of the 1968 original. MR 1163370
  • 9. David Freedman, Brownian motion and diffusion, Holden-Day, San Francisco, Calif.-Cambridge-Amsterdam, 1971. MR 0297016
  • 10. Masatoshi Fukushima, Dirichlet forms and Markov processes, North-Holland Mathematical Library, vol. 23, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1980. MR 569058
  • 11. J. M. Harrison and L. A. Shepp, On skew Brownian motion, Ann. Probab. 9 (1981), no. 2, 309–313. MR 606993
  • 12. H. Hoteit, R. Mose, A. Younes, F. Lehmann, and Ph. Ackerer, Three-dimensional modeling of mass transfer in porous media using the mixed hybrid finite elements and the random-walk methods, Math. Geol. 34 (2002), no. 4, 435–456. MR 1951790,
  • 13. K. Itô and H. P. McKean Jr., Brownian motions on a half line, Illinois J. Math. 7 (1963), 181–231. MR 0154338
  • 14. Olav Kallenberg, Foundations of modern probability, 2nd ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002. MR 1876169
  • 15. Reinhard Lang, Effective conductivity and skew Brownian motion, J. Statist. Phys. 80 (1995), no. 1-2, 125–146. MR 1340556,
  • 16. J.-F. Le Gall, One-dimensional stochastic differential equations involving the local times of the unknown process, Stochastic analysis and applications (Swansea, 1983) Lecture Notes in Math., vol. 1095, Springer, Berlin, 1984, pp. 51–82. MR 777514,
  • 17. Antoine Lejay, On the constructions of the skew Brownian motion, Probab. Surv. 3 (2006), 413–466. MR 2280299,
  • 18. Antoine Lejay and Miguel Martinez, A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients, Ann. Appl. Probab. 16 (2006), no. 1, 107–139. MR 2209338,
  • 19. Zhi Ming Ma and Michael Röckner, Introduction to the theory of (nonsymmetric) Dirichlet forms, Universitext, Springer-Verlag, Berlin, 1992. MR 1214375
  • 20. Petr Mandl, Analytical treatment of one-dimensional Markov processes, Die Grundlehren der mathematischen Wissenschaften, Band 151, Academia Publishing House of the Czechoslovak Academy of Sciences, Prague; Springer-Verlag New York Inc., New York, 1968. MR 0247667
  • 21. Michèle Mastrangelo and Mouloud Talbi, Mouvements browniens asymétriques modifiés en dimension finie et opérateurs différentiels à coefficients discontinus, Probab. Math. Statist. 11 (1990), no. 1, 47–78 (French, with English summary). With the collaboration of Victor Mastrangelo and Youssef Ouknine. MR 1096939
  • 22. Y. Ouknine, Le “Skew-Brownian motion” et les processus qui en dérivent, Teor. Veroyatnost. i Primenen. 35 (1990), no. 1, 173–179 (French); English transl., Theory Probab. Appl. 35 (1990), no. 1, 163–169 (1991). MR 1050069,
  • 23. N. I. Portenko, Generalized diffusion processes, Translations of Mathematical Monographs, vol. 83, American Mathematical Society, Providence, RI, 1990. Translated from the Russian by H. H. McFaden. MR 1104660
  • 24. N. I. Portenko, A probabilistic representation for the solution to one problem of mathematical physics, Ukraïn. Mat. Zh. 52 (2000), no. 9, 1272–1282 (English, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 52 (2000), no. 9, 1457–1469 (2001). MR 1816940,
  • 25. Jorge M. Ramirez, Enrique A. Thomann, Edward C. Waymire, Roy Haggerty, and Brian Wood, A generalized Taylor-Aris formula and skew diffusion, Multiscale Model. Simul. 5 (2006), no. 3, 786–801. MR 2257235,
  • 26. Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1991. MR 1083357
  • 27. R. E. Showalter, Hilbert space methods for partial differential equations, Electronic Monographs in Differential Equations, San Marcos, TX, 1994. Electronic reprint of the 1977 original. MR 1302484
  • 28. Daniel W. Stroock, Diffusion semigroups corresponding to uniformly elliptic divergence form operators, Séminaire de Probabilités, XXII, Lecture Notes in Math., vol. 1321, Springer, Berlin, 1988, pp. 316–347. MR 960535,
  • 29. Daniel W. Stroock and S. R. Srinivasa Varadhan, Multidimensional diffusion processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 233, Springer-Verlag, Berlin-New York, 1979. MR 532498
  • 30. K. Ueno, F. Kitagawa, and N. Kitamura, Photocyanation of pyrene across an oil/water interface in a polymer microchannel chip, Lab on a Chip: Miniaturisation for Chemistry, Biology and Bioengineering 2 (2002), 231-234.
  • 31. J.B. Walsh, A diffusion with a discontinuous local time, Astérisque 52-53 (1978), 37-45.

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Additional Information

Jorge M. Ramirez
Affiliation: Escuela de Matemáticas, Universidad Nacional de Colombia Sede Medellin, Calle 59A, No. 63-20, Medellin, Colombia

Keywords: Skew Brownian motion, diffusion, layered media
Received by editor(s): May 13, 2010
Received by editor(s) in revised form: August 27, 2010
Published electronically: March 3, 2011
Additional Notes: The author’s research was partially supported by The University of Arizona and grants DMS-0073958 and CMG 03-27705 from the National Science Foundation to Oregon State University.
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.